Theory and measurement


 Adela Phelps
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1 Gavity: Theoy and measuement Reading: Today: p11  Theoy of gavity Use two of Newton s laws: 1) Univesal law of gavitation: ) Second law of motion: Gm1m F = F = mg We can combine them to obtain the gavitational acceleation at the suface of the eath: GM g = R E E Is the Eath s gavitational acceleation a constant?
2 Vaiations in g Lage scale vaiations: global o egions Smalle scale vaiations: local This is what we want to make use of The geoid Mean sea level is an equipotential suface it is the geoid
3 Gavity and potentials g is a vecto field: g GM E = whee 1 is the unit vecto pointing 1 towad the cente of the Eath RE Gavitational potential: Gm U = U is a scala field which makes it easie to wok with Definition: The gavitational potential, U, due to a point mass m, at a distance fom m, is the wok done by the gavitational foce in moving a unit mass fom infinity to to a position fom m. Relating g to U U is a scala field which makes it easie to wok with: Potentials ae additive Gavity is a consevative foce And gavitational acceleation can be easily detemined fom the potential Given: It follows that: Gm U = U g = = Gm Fo smalle scale poblems we usually deal with g, and sum the vetical component of g
4 Gavity anomalies Sum contibutions in the vetical diection g z dm ρdv = G cosφ = G M V cosφ O, in Catesian coodinates: g z ρzdxdydz = G 3 whee = ( x α ) + ( y β ) + z This is ideal fo implementation in a compute code. Units fo g SI unit fo g: m/s though you will aely see this! 1 cm/s = 1 Gal (fo Galileo) = 0.01 m/s milligal o mgal = 103 Gal typical unit fo field studies Ou text book uses the gavity unit (g.u.) 1 g.u. = 0.1 mgal Nomal value of g at the suface of the Eath: g E = 9.8 m/s = 980 cm/s = 980 Gal = 980,000 mgal = 9800 g.u.
5 Rock density Mass = Density x Volume Lateal vaiations in ock density esult in gavity anomalies that can be measued at the suface Factos influencing ock density Unconsolidated sediments composition, poosity, satuation Sedimentay ocks composition, age and depth of buial (compaction), cementation, poosity, poe fluid Igneous ocks composition (esp. silica content), cystal size, factuing (i.e. poosity) Metamophic ocks composition (esp. silica content), metamophic gade, factuing (i.e. poosity) Poosity and poe fluid content ae pobably the most impotant factos affecting density in the shallow subsuface
6 Table of ock densities Sedimentay ovebuden Igneous/metamophic basement Similaity in ock densities can make it difficult to distinguish Measuing g: Absolute and elative g at the Eath s suface ~ 980,000 mgal vaiations in g on the ode 1 mgal need to measue g to bette than 1 pat in 1 million use instuments sensitive to elative changes in g
7 Measuing g: Absolute gavity Measuing g: Stable gavimete change in g change in sping length Hooke s Law F = k L and g = k L/m if g/g = 106 then L/L = 106 This equies high optical, mechanical o electonic magnification
8 Measuing g: Unstable gavimete Applies and additional negative estoing foce to amplify changes in g Uses a zeo length sping: the estoing foce is equal to the length of the sping Suitable choice of mass, sping constant and geomety makes the system unstable and vey sensitive to changes in g LaCosteRombeg gavimete Gavity suveying Suvey design s Suvey design consideations Unifom gid fo easie intepetation Station spacing: s < h h is the depth of the body of inteest Avoid steep tomogaphic gadients Absolute and elative station locations ae needed how accuate? Typical station spacing Regional geologic studies: km to 10s of km Local stuctue/engineeing/envionmental: 10s to 100s m Nea suface e.g. acheology: few metes
9 Gavity suveying Dift The eading of a gavimetes at a point changes with time! Causes Instument dift: due to envionmental changes (P,T) and sping ceep Eath tides: elative otations of the eath, moon and sun Gavity suveying Coecting fo dift 1. Retun to base station peiodically. Assume dift is linea 3. Coect measuements in loop How often? Depends on equies accuacy max tidal ate: 0.05 mgal/h instument dift usually less
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